Free electron lasers are able to produce very high power laser radiation in an efficient manner. In addition to their high efficiency and high power capability, they are attractive since they can be tunable over a wide spectrum from millimeter wavelengths to the x-ray region. Free electron lasers pass a relativistic electron beam through a spatially varying magnetic field called a wiggler, which wiggles the electrons in the electron beam. The wiggle of the electrons cause the electrons to radiate. If the proper phase is maintained by the electron beam, the radiation produced can amplify an existing electromagnetic field creating a laser beam. Tuning the wiggler so that the electrons emit light of a desired wavelength and so that the light can be efficiently extracted from the electron beam, comprises adjusting the magnetic field strength in the wiggler.
There are other applications for wigglers, such as third generation synchrotron radiation rings. These synchrotron radiation rings employ wigglers to increase radiation power output and expand or customize the wavelength spectrum of the radiation produced. In the claims and specification the word "wiggler" will also include those wigglers called undulators producing narrowly peaked radiation spectrums.
FIG. 1 is a schematic drawing of a relativistic electron beam passing through a wiggler to produce directed radiation. Alternating magnetic poles 12 are used to create an alternating magnetic field B.sub.w, which in the x-z plane is parallel and antiparallel to the y axis. An electron beam 10 is directed along the z axis through the alternating magnetic field. The magnetic field along the z axis is known as the on-axis magnetic field. The maximum magnetic field values along the z-axis defines the magnitude of B.sub.w The movement along the z direction through an alternating magnetic field causes the electron beam 10 to oscillate in the x direction causing a sinusoidal path 14. The oscillation of the relativistic electrons creates electromagnetic radiation 6 with a frequency which is a function of the electron energy, the oscillation frequency, and the magnetic field strength. The electromagnetic radiation is highly directional in the z direction.
FIG. 2 is a cut away view of a wiggler used in the prior art. An upper magnetic yoke 16 is used to hold a plurality of upper magnetic poles with a first upper magnetic pole 18. A lower magnetic yoke 20 is used to hold a plurality of lower magnetic poles with a first lower magnetic pole 22. The magnetic poles and the magnetic yokes 16, 20 are made of a ferromagnetic material. Adjacent to the first upper magnetic pole are two permanent magnets 24. Adjacent to the first lower magnetic pole are two permanent magnets 26.
The first upper magnetic pole 18 and the adjacent magnets 24 are used as a mandrel for a first upper electromagnetic coil 36. The first lower magnetic pole 22 and the adjacent magnets 26 are used as a mandrel for a first lower electromagnetic coil 38.
The first upper electromagnetic coil 36 and the first lower electromagnetic coil 38 are wound so that as viewed from above looking down in the -y direction the currents in the first upper electromagnetic coil 36 and the first lower electromagnetic coil 38 flow in a clockwise direction. Current in the clockwise direction in the first upper electromagnetic coil 36 creates in the first upper magnetic pole 18 a flux in the downward (-y) direction from the upper magnetic yoke 16 to the tip 28 of the first upper magnetic pole 18. Current in the clockwise direction in the first lower electromagnetic coil 38 creates in the first lower magnetic pole 22 a flux in the downward (-y) direction from the tip 30 of the first lower magnetic pole 22 to the lower magnetic yoke 20. This results in a net effect of a magnetic flux in a downward direction passing across the gap between the first upper magnetic pole 18 and the first lower magnetic pole 22.
The two permanent magnets 24 adjacent to the first upper magnetic pole 18 are oriented to place in the first upper magnetic pole 18 a net magnetic flux into the pole at the location of the first upper adjacent permanent magnets 24. This flux then travels in the general upward (+y) direction in the pole toward the upper magnetic yoke 16. The two permanent magnets 26 adjacent to the first lower magnetic pole 22 are oriented to place in the first lower magnetic pole 22 a net magnetic flux out of the pole at the location of the first lower adjacent permanent magnets 26. This flux travels in the general upward (+y) direction in the pole from the lower magnetic yoke 20. The magnetic flux from the permanent magnets is induced in the poles but does not cross the gap between the first upper magnetic pole 18 and the first lower magnetic pole 22.
A second upper pole has two adjacent permanent magnets 32. The second lower pole has two adjacent permanent magnets 34. The second upper magnetic pole and the adjacent magnets 32 are used as a mandrel for a second upper electromagnetic coil 35. The second lower magnetic pole and the adjacent magnets 34 are used as a mandrel for a second lower electromagnetic coil 37.
The second upper electromagnetic coil 35 and the second lower electromagnetic coil 37 are wound so that as viewed from above looking down in the -y direction the current in the second upper electromagnetic coil 35 and the second lower electromagnetic coil 37 flow in a counterclockwise direction. Current in the counterclockwise direction in the second upper electromagnetic coil 35 and current in the counterclockwise direction in the second lower electromagnetic coil 37 create a flux in the upward (+y) direction. This results in a net effect of a magnetic flux in a upward direction passing across the gap between the second upper magnetic pole and the second lower magnetic pole.
The two permanent magnets 32 adjacent to the second upper magnetic pole are oriented to place in the second upper magnetic pole a magnetic flux in the downward (-y) direction. The two permanent magnets 34 adjacent to the second lower magnetic pole are oriented to place in the second lower magnetic pole a magnetic flux in the downward (-y) direction. The magnetic flux from the permanent magnets is induced in the poles and does not cross the gap between the second upper magnetic pole and the second lower magnetic pole.
Each pole and adjacent set of permanent magnet are used as a mandrel for an electromagnetic coil. The electromagnetic coils are used to induce a magnetic flux in the magnetic poles and thus in the gap between opposite poles giving rise to the alternating magnetic field which causes the electron beam spatial oscillations or "wiggles." Adjusting the current in the electromagnetic coil changes the magnitude of the magnetic field and thus allows the tuning of the wiggler to either 1) compensate for a decrease in electron beam energy along the z direction and thus maintain a resonance condition between the electron beam and the radiation being amplified over a larger spatial distance or 2) change the frequency of the electromagnetic radiation produced by an electron beam of a given energy passing between the tips of the magnetic poles of the wiggler.
Steering coils 40 are wrapped around the upper magnetic yoke 16. The steering coils 40 provide a magnetic field used to make minor steering corrections of the electron beam as it passes through the wiggler.
FIG. 3 is a graph of a hysteresis loop for an iron material. The magnetizing force H applied to the iron material is plotted along the abscissa, and the magnetic induction B induced in the iron is plotted along the ordinate. The slope of the curve forming the loop at a point on the curve is .mu.=.mu..sub.o .mu..sub.r, where .mu..sub.o is the free-space permeability. At B=0, for some iron .mu..sub.r =1,000. At point b, .mu..sub.r is close to one. At point b, the iron is magnetically saturated. At point b, an increase in the magnetizing force H, causes only a slight increase in the induced magnetism B in the iron. At points c and d, one side of the hysteresis loop goes from being approximately linear at B=0 to becoming significantly nonlinear. For a wiggler made of this iron, the sum of the magnetic fields in the iron pole induced by the permanent magnets and the electromagnetic coil around a pole is kept between H.sub.d and H.sub.c. This is practiced for two reasons. Beyond H.sub.c and H.sub.d the absolute value of .mu. decreases, decreasing the change in B for a unit change in H, thus making the change in H less efficient outside of the range. Secondly, the slope .mu. becomes variable, making B harder to predict outside of the range. In the claims and specification, applying summed magnetic fields in the pole outside of the range H.sub.d to H.sub.c will be considered a saturating magnetic flux density in the poles.
The design of an iron-core electromagnetic wiggler pole is largely an exercise in simultaneously sufficiently limiting both the maximum magnetic flux density in the iron of the pole structure and the current density in the electromagnetic coils while satisfying system level requirements, minimizing cost and technical risks, etc. A wiggler must often attain the following three systems level goals: (1) high wiggler on-axis magnetic flux density (magnetic field), (2) low magnetic field errors (including those due to saturation of the poles), and (3) widely tunable range. Wiggler design features enabling the attainment of the first goal, e.g. larger electromagnetic coil currents and/or cross-sectional areas, tend to inhibit the attainment of the second due to the onset of magnetic saturation of the wiggler poles. Wiggler pole magnetic saturation also inherently limits the degree to which the first goal can be attained due to the leveling off of the slope beyond the saturation points. K. Halbach in "Some Concepts To Improve The Performance Of DC Electromagnetic Wigglers," Nuclear Instruments and Methods in Physics Research A250 (1986) pp 115-119, North-Holland, Amsterdam describes the design which enables the attainment of much higher magnetic flux densities (while also maintaining a low level of magnetic field errors) in electromagnet wigglers by employing permanent magnets 24, 26, 32, 34 to put a reverse bias magnetic flux in the wiggler pole, without directly altering the wiggler's on-axis magnetic field. This allows the electromagnetic coil current (and thus on-axis magnetic flux density) to be increased to a higher level before the onset of wiggler pole magnetic saturation.
FIG. 4a is a cross section of half a pole shown in FIG. 2 along cut lines 4--4 with a graph of the magnetic flux density along the pole. The magnitude of the on-axis magnetic field B.sub.w is proportional to the magnetic scalar potential at the tip 28 of the pole (U(T)), so that EQU B.sub.w =qU(T), (1)
where q is a constant. U(T) is proportional to the number of ampere-turns in the electromagnetic coil surrounding the coil. The scalar potential anywhere along the pole is given by EQU U(y)=U(T)(1-(y-T)/h), (2)
where T is the value of y at the tip of the pole and h is the height of the electromagnet coil 36. The increment in electromagnet coil 36 induced magnetic flux entering (or leaving) the pole per unit vertical distance along the pole (.delta..PHI..sub.EM /.delta.y) is proportional (to first order) to the magnetic scalar potential at that location on the pole (U(y)). Thus from an electromagnetic coil, the induced magnetic flux in the pole which it surrounds is: ##EQU1## where c is a constant and h is the height of the electromagnetic coils. Since .PHI..sub.EM (T) is proportional to U(T), EQU .PHI..sub.EM (y)=U(T)(k+c((y-T)-(y-T).sup.2 /2h), (3)
where k is a constant. .PHI..sub.EM (y) is maximum at y=y.sub.base =T+h and has a value: EQU .PHI..sub.EMmax =.PHI..sub.EM (T+h)=U(T)(k+c(h/2))=cU(T)(k/c+h/2). (4)
Thus, the magnetic flux density in the pole 18 is a function of both the number of ampere-turns in the electromagnetic coil 36 and the location of those ampere-turns in the electromagnetic coil 36 on the pole 18, while the on-axis magnetic flux density is a function to first order) of the number of ampere-turns only, irrespective of their location in the electromagnetic coil 36 along the pole 18. Equations 1-4 are depicted graphically in FIG. 4. With cU(y) plotted along the abscissa and y along the ordinate, the slope 41 of the shaded region shows how the scalar potential U varies as a function of y according to equation 2. Since B.sub.w is proportional to U(T), the on-axis magnetic flux density is proportional to the width of the base of the shaded region 42, and since B.sub.pole is proportional to .PHI..sub.pole, the electromagnet-induced pole magnetic flux density at any given y is proportional to the area of that portion of the shaded region 42 below that y according to equation 3. In particular, at the base, the maximum electromagnet-induced magnetic flux density is proportional to the area of the entire shaded region, according to equation 4. For simplicity, the proportionality constant between the area of the shaded region 42 and the maximum electromagnet induced pole magnetic flux density is set to 1 in the following examples.
In an example of the requirements for certain iron wigglers the iron reaches its saturation point at .+-.14 kilo Gauss (kG). To avoid the saturation range, the absolute value of the sum of the magnetic flux density of the electromagnetic coil in the pole (EM) and the magnetic flux density of the permanent magnet in the pole (PM) everywhere within the pole must be less than or equal to 14 kG, denoted by the equation: -14.ltoreq.(EM+PM).ltoreq.14. In this example the permanent magnet induced a magnetic flux density at the base of the pole is -20 kG. Then to avoid saturation, the magnetic flux density at the base of the pole from the electromagnetic coil EM must fall in the range 6&lt;EM&lt;34. This means that the shaded region corresponding to the electromagnetic coil induced magnetic flux density at the pole base must have an area between 6 and 34. FIGS. 4b, c illustrate the range of on-axis magnetic flux densities and the corresponding range of the magnetic flux densities in the pole. In FIG. 4b the saturation limited maximum pole flux density, and thus the pole tip potential corresponding to the saturation-limited maximum on-axis magnetic flux density are shown. The distances over which the ampere-turns are applied is from T=2 to y.sub.base =6 so that h=4. Since the area of the shaded region is set equal to the maximum flux density in the pole, the area of the shaded region is 34 kG. Using the equation for the area of a triangle A=HB/2 and the equation for the area of a rectangle A=HB, where H refers to the height and B refers to the base of the triangle or rectangle. 34=4cU(T)/2+2cU(T). Therefore, cU(T)=8.5 as denoted along the abscissa. The value B.sub.w is proportional to the scalar potential at the tip (c.f. equation 1), and in this configuration, B.sub.w, max =8.5(q/c) is the maximum on-axis magnetic flux density (represented by the width of the base of the shaded region), given the maximum magnetic flux density of 34 (represented by the area of the shaded region) and the geometry of the pole and electromagnet.
In FIG. 4c the saturation limited minimum pole flux density and thus the pole tip potential corresponding to the saturation-limited minimum on-axis magnetic flux density are shown. Since the area of the shaded region is set equal to the minimum flux density in the pole, the area of the shaded region is 6 kG. From the equation for the area of the shaded region 6=4cU(T)/2+2cU(T). Therefore, cU(T)=1.5, as denoted along the abscissa. In this configuration, B.sub.w, min =1.5(q/c) (represented by the width of the base of the shaded region) is the minimum on-axis magnetic flux density value, given the minimum pole electromagnet magnetic flux density of 6 (represented by the area of the shaded region) and the geometry of the pole and electromagnet. By decreasing the height h of the electromagnet coils, B.sub.w, max can be increased for a given B.sub.pole, max but then B.sub.w, min would also be increased. By increasing h, B.sub.w, min can be decreased for a given B.sub.pole, min but then B.sub.w, max would also be decreased. What would be desirable is a means to both increase B.sub.w, max and decrease B.sub.w, min thus increasing the tunable range of the wiggler.
It should be noted that the slopes 41 of the boundary of the shaded region in FIGS. 4abc are proportional to .DELTA.y/.DELTA.U. Heat transfer limitations restrict the maximum allowable current density in the electromagnet coils. The coil current density, J.sub.coil is proportional to the rate of change of the magnetic scalar potential along the pole face, dU/dy. Therefore the heat transfer constraint limiting the magnitude of J.sub.coil effectively puts a lower bound on the slope 41 of the boundary of the shaded region. A vertical boundary (infinite slope) implies a zero current in the coil (as in FIG. 10 for example), while a physically impossible horizontal boundary would imply an infinite current in the coil. The important point is that there are two design constraints: a magnetic saturation constraint and a heat transfer contraint. In FIG. 4b, then, maximum B.sub.w is attained by increasing coil current until either (1) the pole saturates (i.e. the area of the shaded region 42 is 34) or (2) the slope 41 reaches its heat transfer limited maximum allowable value, whichever comes first. We have assumed this example is saturation limited.
The base of the pole is the location where the electromagnets induce the greatest flux density in the pole. When the permanent magnets are used to apply a reverse bias flux in the pole, then the permanent magnet may induce an incremental flux in the pole at a specific location that exceeds the incremental flux induced by the electromagnet at that location and it is possible that pole saturation may first occur at a location other than the pole base. Thus, in general, one must insure that -14.ltoreq.EM+PM.ltoreq.14 for all pole locations "y". The illustrative example of FIG. 4 assumed the pole first saturated at the base, however the general principles for determining the tuning range, outlined above, are not restricted to this special case.
FIG. 5 illustrates the change in the tunable range caused by adding the adjacent permanent magnets to the poles. The dashed line 141 indicates the range over which electromagnetic coils alone may induce flux into the pole without incurring saturation. The solid line 142 indicates the range over which electromagnetic coils may induce flux into a pole surrounded by adjacent reverse-biasing permanent magnets. The adjacent permanent magnets shift the range but the width of the range due to the electromagnets for a pole surrounded by adjacent permanent magnets 44 remains approximately equal to the width of the range due to the electromagnets alone 43.
FIG. 6 illustrates another type of wiggler assembly used in the prior art as described by K. Halbach in "Some Concepts To Improve The Performance of DC Electromagnetic Wigglers" cited above. The apparatus shown here uses sheets of permanent magnets (laced magnets) 45 between the electromagnetic coil windings 46 in addition to the permanent magnets 48 adjacent to the poles 47 to further increase the attainable on-axis magnetic flux density beyond that attainable with the use of only adjacent permanent magnets. Since the electromagnetic flux density in the pole is correspondingly increased for the same amount of current in the electromagnet due to its vertical displacement on the pole so as to accommodate the laced permanent magnet, the minimum attainable on-axis magnetic flux density B.sub.w, min increases more than does the maximum on-axis magnetic flux density B.sub.w, max and thus the tunable range decreases. In the prior art the plurality of electromagnetic coils surrounding a pole were electrically connected so that the current through each coil along a pole is not independently controlled. It would be desirable to increase the tunable range of this apparatus.